Beat together sugar and eggs in a large bowl with an electric mixer at medium speed just until combined, about 1 minute. Reduce speed to low and add flour and milk mixtures alternately in batches, beginning and ending with flour mixture and mixing until just combined.

Divide batter evenly between cake pans, smoothing tops, and bake until a wooden pick or skewer inserted in center of each cake layer comes out clean, 35 to 40 minutes.

Cool cake layers in pans on racks 10 minutes. Run a thin knife around edge of 1 cake layer and invert rack over cake pan, then invert cake onto rack. Repeat with second layer. Peel off paper and cool layers completely.

Make curd:
Whisk together zest, lemon juice, sugar, yolks, a pinch of salt, and guar gum in a 1-quart heavy saucepan. Add butter and cook over moderately low heat, whisking constantly, until curd is thick enough to hold marks of whisk and first bubbles appear on surface, about 5 minutes. Whisk in extract. Immediately pour curd into a bowl, then cover surface with wax paper and chill until cold, about 30 minutes.

Make frosting:
Beat butter with an electric mixer at high speed until light and fluffy, about 1 minute. Reduce speed to low and add confectioners sugar, lemon juice, extract, and zest, then mix until creamy and smooth, about 2 minutes.

Frost cake:
Halve each cake layer horizontally using a long serrated knife. Spread bottom half of each cake layer with half of lemon curd, then top with remaining cake layers to form two sandwiched cakes. Put 1 sandwiched cake on a cake stand or platter and spread a heaping 1/2 cup frosting over top, then cover with remaining sandwiched cake. Frost top and sides of cake with remaining frosting.

Cooks' notes:

 Cake layers can be made (but not halved) 1 day ahead and cooled completely, then kept, wrapped well in plastic wrap, at room temperature.
 Lemon curd can be chilled up to 3 days.
 Cake can be completely assembled and frosted 4 hours ahead and kept at room temperature, or 1 day ahead and chilled, loosely covered. Bring to room temperature before serving.